Question
Answer
$$(x-(3+i))^2(x-(3-i))^2=i^4-2i^2x^2+x^4+12i^2x-12x^3-18i^2+54x^2-108x+81$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-(3+i))^2(x-(3-i))^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x-3-i)^2(x-3+i)^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1i^2-2ix+x^2+6i-6x+9)(1i^2+2ix+x^2-6i-6x+9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}i^4-2i^2x^2+x^4+12i^2x-12x^3-18i^2+54x^2-108x+81\end{aligned} $$ | |
| ① | Remove the parentheses by changing the sign of each term within them. $$ - \left( 3+i \right) = -3-i $$Remove the parentheses by changing the sign of each term within them. $$ - \left( 3-i \right) = -3+i $$ |
| ② | Multiply each term of $ \left( \color{blue}{x-3-i}\right) $ by each term in $ \left( x-3-i\right) $. $$ \left( \color{blue}{x-3-i}\right) \cdot \left( x-3-i\right) = x^2-3x-ix-3x+9+3i-ix+3i+i^2 $$ |
| ③ | Combine like terms: $$ x^2 \color{blue}{-3x} \color{red}{-ix} \color{blue}{-3x} +9+ \color{green}{3i} \color{red}{-ix} + \color{green}{3i} +i^2 = i^2 \color{red}{-2ix} +x^2+ \color{green}{6i} \color{blue}{-6x} +9 $$Multiply each term of $ \left( \color{blue}{x-3+i}\right) $ by each term in $ \left( x-3+i\right) $. $$ \left( \color{blue}{x-3+i}\right) \cdot \left( x-3+i\right) = x^2-3x+ix-3x+9-3i+ix-3i+i^2 $$ |
| ④ | Combine like terms: $$ x^2 \color{blue}{-3x} + \color{red}{ix} \color{blue}{-3x} +9 \color{green}{-3i} + \color{red}{ix} \color{green}{-3i} +i^2 = i^2+ \color{red}{2ix} +x^2 \color{green}{-6i} \color{blue}{-6x} +9 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{i^2-2ix+x^2+6i-6x+9}\right) $ by each term in $ \left( i^2+2ix+x^2-6i-6x+9\right) $. $$ \left( \color{blue}{i^2-2ix+x^2+6i-6x+9}\right) \cdot \left( i^2+2ix+x^2-6i-6x+9\right) = \\ = i^4+ \cancel{2i^3x}+i^2x^2 -\cancel{6i^3}-6i^2x+9i^2 -\cancel{2i^3x}-4i^2x^2 -\cancel{2ix^3}+12i^2x+ \cancel{12ix^2} -\cancel{18ix}+i^2x^2+ \cancel{2ix^3}+x^4 -\cancel{6ix^2}-6x^3+9x^2+ \cancel{6i^3}+12i^2x+ \cancel{6ix^2}-36i^2 -\cancel{36ix}+ \cancel{54i}-6i^2x -\cancel{12ix^2}-6x^3+ \cancel{36ix}+36x^2-54x+9i^2+ \cancel{18ix}+9x^2 -\cancel{54i}-54x+81 $$ |
| ⑥ | Combine like terms: $$ \text{ \text{Text Is Too Long To Display} } = \\ = i^4 \color{orange}{-2i^2x^2} +x^4+ \color{orange}{12i^2x} \color{red}{-12x^3} \color{red}{-18i^2} + \color{orange}{54x^2} \color{blue}{-108x} +81 $$ |
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